ACT Math Triangles and Circle Theorems Quiz

The circumference of a circle is calculated using the formula \( C = 2\pi r \), where \( r \) is the radius. With a radius of 6 cm, the calculation becomes \( C = 2\pi \times 6 = 12\pi \) cm. Thus, the circumference of the circle is 12π cm.

An inscribed angle in a circle is half the measure of the central angle that subtends the same arc. Given that the inscribed angle measures 40°, the central angle can be calculated by doubling this measure. Therefore, the central angle is 2 × 40° = 80°.

In isosceles triangle DEF, angles opposite equal sides are equal. Since DE = DF, angles E and F are equal. Given angle E is 55°, angle F must also be 55° to maintain the properties of the isosceles triangle. Thus, angle F is 55°.

A tangent line touches a circle at exactly one point, known as the point of tangency. At this point, the radius drawn to the point of tangency forms a right angle with the tangent line, demonstrating that the tangent is perpendicular to the radius. This geometric property is fundamental in circle geometry.

The angle formed by the intersecting chords is calculated using the formula: angle = (arc1 + arc2) / 2. Here, the arcs measure 60° and 80°. Adding these gives 140°, and dividing by 2 results in an angle of 70°, which represents the angle formed by the intersecting chords inside the circle.

In any triangle, the sum of the three interior angles always equals 180 degrees. This is a fundamental property of Euclidean geometry, derived from the fact that a triangle can be divided into two right triangles, each contributing to the total angle measure. This relationship holds true regardless of the triangle's shape or size.

Triangle GHI is a right triangle because it satisfies the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Here, 5² + 12² equals 13², confirming it is a right triangle.

To find the area of a circle, use the formula A = πr². The radius is half the diameter, so with a diameter of 14 units, the radius is 7 units. Squaring the radius gives 49, and multiplying by π results in an area of 49π square units.

In triangle JKL, with a right angle at J, we can apply the Pythagorean theorem: \( a^2 + b^2 = c^2 \). Here, \( a = JK = 3 \) and \( b = JL = 4 \). Calculating, \( 3^2 + 4^2 = 9 + 16 = 25 \), so \( c = \sqrt{25} = 5 \). Thus, the hypotenuse KL is 5.

When two secants are drawn from an external point to a circle, the angle formed at that point is equal to half the difference of the measures of the intercepted arcs. This relationship stems from the properties of angles formed by secants and arcs in circle geometry, highlighting the connection between angles and arc measures.

In an equilateral triangle, all sides are of equal length and all interior angles are equal. Since the sum of the interior angles in any triangle is 180°, each angle in an equilateral triangle measures 180° divided by 3, resulting in 60°. Hence, each interior angle measures 60°.

In a triangle, the sum of all angles is always 180°. Given angles A and B measure 45° and 65° respectively, we can calculate angle C by subtracting the sum of angles A and B from 180°. Thus, angle C = 180° - (45° + 65°) = 70°.